2026/01/27 by Antonio Agresti, Fabian Germ, Mark Veraar
#math.PR #math.AP #math.FA
In the study of stochastic PDEs with colored, non-trace class space-time noise, one frequently encounters Gaussian series of the form g ∑n≥ 1 γn μn fn, where (γn)n is a sequence of standard independent Gaussian variables, g is an Lη(O) function, (μn)n is a sequence of scalars, and (fn)n is an orthonormal system in L2(O) where O ⊆ ℝd is an open set. In this manuscript, we establish necessary and sufficient conditions for the above sum to converge in Bessel potential spaces H-s,q(O). The latter can be interpreted as a Sobolev embedding for Gaussian series. Our main theorem is formulated using weighted sequence spaces that encode the L^∞-growth of the orthonormal system (fn)n, a feature that is crucial for obtaining sharp estimates. We apply our results to the stochastic heat equation with additive non-trace class noise. In this case, our conditions capture the scaling relationship between the heat operator and the coloring of the noise.